Question

What is `(dy)/(dx)`of `xsqrt(1+y)+ysqrt(1+x)=0??`

Answer 1

`y' = -(sqrt(1+y) +y/(2sqrt(1+x)))/(x/(2sqrt(1+y)) + sqrt(1+x))`

Explanation:

Given: `x sqrt(1+y) + y sqrt(1+x) = 0`

We need to use implicit differentiation. We will use these differentiation rules for each segment of the equation:

The product rule: `(u*v)' = u*v' + v*u'`

The power rule: `(u^n)' = n u^(n-1) u'`

`(0)' = 0`

First segment of the equation:
Let `u = x; " "u' = 1`
`" "v = sqrt(1+y) = (1+y)^(1/2); " " v' = 1/2(1+y)^(-1/2)y'`
`" "v' = (y')/(2sqrt(1+y)`

Second segment of the equation:
Let `u = y; " "u' = y'`
`" "v = sqrt(1+x) = (1+x)^(1/2); " " v' = 1/2(1+x)^(-1/2)(1)`
`" "v' = 1/(2sqrt(1+x)`

Putting it all together using the product rule:
`(xy')/(2sqrt(1+y)) + sqrt(1+y) + y/(2sqrt(1+x)) +y' sqrt(1+x)= 0`

Keep all terms that contain `y'` on the left side of the equation:
`(xy')/(2sqrt(1+y)) +y' sqrt(1+x) = -sqrt(1+y) -y/(2sqrt(1+x))`

Factor `y'`;

`y' (x/(2sqrt(1+y)) + sqrt(1+x)) = -(sqrt(1+y) +y/(2sqrt(1+x)))`

Isolate `y'` using division:

`y' = -(sqrt(1+y) +y/(2sqrt(1+x)))/(x/(2sqrt(1+y)) + sqrt(1+x))`

This answer can be simplified if needed by finding common denominators in both the numerator and denominator.